We have a triangle with sides a, b, c that ab+bc+ca=12
Prove that 6=<a+b+c<7
I proved that a+b+c is no less than 6 but how to prove that it's less than 7?
We have a triangle with sides a, b, c that ab+bc+ca=12
Prove that 6=<a+b+c<7
I proved that a+b+c is no less than 6 but how to prove that it's less than 7?
a^{2}+b^{2} >= 2ab
a^{2}+c^{2} >= 2ac
b^{2}+c^{2} >= 2bc
a^{2}+b^{2}+c^{2} >= ab+ac+bc
a^{2}+b^{2}+c^{2}+2ab+2ac+2bc >= 3ab+3ac+3bc(a+b+c)^{2 }>= 36
a+b+c >=6
Last edited by eligotas; 01-15-2019 at 10:05 AM.
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